Term Rewriting System R:
[x, y, z]
app(app(app(f, 0), 1), x) -> app(app(app(f, x), x), x)
app(app(app(f, x), y), z) -> 2
app(app(app(g, x), x), y) -> y
app(app(app(g, x), y), y) -> x
0 -> 2
1 -> 2
Innermost Termination of R to be shown.
R
↳Removing Redundant Rules for Innermost Termination
Removing the following rules from R which left hand sides contain non normal subterms
app(app(app(f, 0), 1), x) -> app(app(app(f, x), x), x)
R
↳RRRI
→TRS2
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
app(app(app(f, x), y), z) -> 2
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g) | = 0 |
| POL(1) | = 0 |
| POL(2) | = 0 |
| POL(app(x1, x2)) | = x1 + x2 |
| POL(f) | = 1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
app(app(app(g, x), y), y) -> x
app(app(app(g, x), x), y) -> y
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g) | = 1 |
| POL(1) | = 0 |
| POL(2) | = 0 |
| POL(app(x1, x2)) | = x1 + x2 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
1 -> 2
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(1) | = 1 |
| POL(2) | = 0 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
0 -> 2
where the Polynomial interpretation:
was used.
All Rules of R can be deleted.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS6
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes